I fear this is an overly mystical misinterpretation of neural nets. (Standard feed-forward) neural nets are layers of non-linear feature transformations. They take inputs and at each step transform those inputs into a more compact representation, distilling the inputs into their most important factors (and throwing away unimportant factors).
So the most likely explanation is that policy/value nets in AlphaGo have learned to extract - with cold logic - the key factors that make up what humans believe to be "good" board positions.
It has little to do with voodoo about neural connections and magic emerging from the weights. AlphaGo has most likely managed to identify the important factors of good board positions (by seeing tons of examples of good and bad moves/positions). It only appears to be magical because these factors are most likely very complex and inter-dependent.
This is supported by the AlphaGo paper - they report that AlphaGo without tree search is about as good as the best tree search programs (amateur pro level). So AlphaGo has taken amateur-pro-level board analysis ability, and combined it with tree search, to achieve top-player performance..